# 3.1 | The Derivative and the Tangent Line Problem

If $$f$$ is defined on an open interval containing $$c$$, and if the limit
$$\lim_{\Delta x \rightarrow 0} \frac{\Delta y}{\Delta x} = \lim_{x \rightarrow 0} \frac{f(c+x)-f(c)}{\Delta x} = m$$
exists, then the line passing through $$(c,f(c))$$ with slope $$m$$ is the tangent line to the graph of $$f$$ at the point $$(c, f(c))$$.
The derivative of $$f$$ at $$x$$ is
$$f'(x) = \lim_{\Delta x \rightarrow 0} \frac{f(x+\Delta x) – f(x)}{\Delta x}$$
provided the limit exists. For all $$x$$ for which this limit exists, $$f’$$ is a function of $$x$$.
If $$f$$ is differentiable at $$x = c$$, then $$f$$ is continuous at $$x = c$$.